# Representation from Young tabloids

I am following the note Young Tableaux and the Representations of the Symmetric Group to work out a representation from a Young tableau for $S_n$.

Here $\lambda$ is a partition of an integer $n$. In that case, the note says following about the basis of the representation.

I would like to compute the matrix for $(3, 1) \in S_4$. How do I work out the basis vectors from the Young tabloids?

The Young tabloids are the basis of $M^\lambda$. So represent $\{t_i\}$ as the standard basis vectors in $\Bbb{C}^4$. That is, $$\{t_1\} = \left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right), \{t_2\} = \left(\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array}\right),$$ and so on... As it says in the notes, the action of $\mathrm{Sym}(4)$ just permutes the subscripts: $\pi \{t_i\} = \{t_{\pi(i)}\}$. As you can see, the permutation $(1,3)$ just swaps $\{t_1\}$ and $\{t_3\}$, and leaves the rest invariant, so the matrix for $(1,3)$ in this basis looks like:

$$(1,3) = \left(\begin{array}{cccc} 0&0&1&0 \\ 0&1&0&0 \\ 1&0&0&0 \\ 0&0&0&1 \end{array}\right)$$